Question 146634
I'll do one of each to get you started


1) 


A)




{{{x + 7y =5}}} Start with the given equation.



{{{y=-(1/7)x+5/7}}} Solve for y.



We can see that the equation {{{y=-(1/7)x+5/7}}} has a slope {{{m=-1/7}}} and a y-intercept {{{b=5/7}}}.



Since parallel lines have equal slopes, this means that we know that the slope of the unknown parallel line is {{{m=-1/7}}}.

Now let's use the point slope formula to find the equation of the parallel line by plugging in the slope {{{m=-1/7}}}  and the coordinates of the given point *[Tex \LARGE \left\(5,7\right\)].



{{{y-y[1]=m(x-x[1])}}} Start with the point slope formula



{{{y-7=(-1/7)(x-5)}}} Plug in {{{m=-1/7}}}, {{{x[1]=5}}}, and {{{y[1]=7}}}



{{{y-7=(-1/7)x+(-1/7)(-5)}}} Distribute



{{{y-7=(-1/7)x+5/7}}} Multiply



{{{y=(-1/7)x+5/7+7}}} Add 7 to both sides. 



{{{y=(-1/7)x+54/7}}} Combine like terms. note: If you need help with fractions, check out this <a href="http://www.algebra.com/algebra/homework/NumericFractions/fractions-solver.solver">solver</a>.



So the equation of the line parallel to {{{x + 7y =5}}} that goes through the point *[Tex \LARGE \left\(5,7\right\)] is {{{y=(-1/7)x+54/7}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,-(1/7)x+5/7,(-1/7)x+54/7),
circle(5,7,0.08),
circle(5,7,0.10),
circle(5,7,0.12))}}}Graph of the original equation {{{y=-(1/7)x+5/7}}} (red) and the parallel line {{{y=(-1/7)x+54/7}}} (green) through the point *[Tex \LARGE \left\(5,7\right\)]. 




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2)


A)





{{{8x + y = 9}}} Start with the given equation.



{{{y=-8*x+9}}} Solve for y.



We can see that the equation {{{y=-8*x+9}}} has a slope {{{m=-8}}} and a y-intercept {{{b=9}}}.



Now to find the slope of the perpendicular line, simply flip the slope {{{m=-8}}} to get {{{m=-1/8}}}. Now change the sign to get {{{m=1/8}}}. So the perpendicular slope is {{{m=1/8}}}.



Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope {{{m=-8}}} and the coordinates of the given point *[Tex \LARGE \left\(3,6\right\)].



{{{y-y[1]=m(x-x[1])}}} Start with the point slope formula



{{{y-6=(1/8)(x-3)}}} Plug in {{{m=1/8}}}, {{{x[1]=3}}}, and {{{y[1]=6}}}



{{{y-6=(1/8)x+(1/8)(-3)}}} Distribute



{{{y-6=(1/8)x-3/8}}} Multiply



{{{y=(1/8)x-3/8+6}}} Add 6 to both sides. 



{{{y=(1/8)x+45/8}}} Combine like terms. note: If you need help with fractions, check out this <a href="http://www.algebra.com/algebra/homework/NumericFractions/fractions-solver.solver">solver</a>.



So the equation of the line perpendicular to {{{8x + y = 9}}} that goes through the point *[Tex \LARGE \left\(3,6\right\)] is {{{y=(1/8)x+45/8}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,-8*x+9,(1/8)x+45/8)
circle(3,6,0.08),
circle(3,6,0.10),
circle(3,6,0.12))}}}Graph of the original equation {{{y=-8*x+9}}} (red) and the perpendicular line {{{y=(1/8)x+45/8}}} (green) through the point *[Tex \LARGE \left\(3,6\right\)].